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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Images of the Brownian sheet


Authors: Davar Khoshnevisan and Yimin Xiao
Journal: Trans. Amer. Math. Soc. 359 (2007), 3125-3151
MSC (2000): Primary 60G15, 60G17, 28A80
DOI: https://doi.org/10.1090/S0002-9947-07-04073-1
Published electronically: February 14, 2007
MathSciNet review: 2299449
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Abstract: An $N$-parameter Brownian sheet in $\mathbf {R}^d$ maps a non-random compact set $F$ in $\mathbf {R}^N_+$ to the random compact set $B(F)$ in $\mathbf {R}^d$. We prove two results on the image-set $B(F)$: (1) It has positive $d$-dimensional Lebesgue measure if and only if $F$ has positive $\frac d 2$-dimensional capacity. This generalizes greatly the earlier works of J. Hawkes (1977), J.-P. Kahane (1985), and Khoshnevisan (1999). (2) If $\dim _{_\mathcal {H}}F > \frac d 2$, then with probability one, we can find a finite number of points $\zeta _1,\ldots ,\zeta _m\in \mathbf {R}^d$ such that for any rotation matrix $\theta$ that leaves $F$ in $\mathbf {R}^N_+$, one of the $\zeta _i$’s is interior to $B(\theta F)$. In particular, $B(F)$ has interior-points a.s. This verifies a conjecture of T. S. Mountford (1989). This paper contains two novel ideas: To prove (1), we introduce and analyze a family of bridged sheets. Item (2) is proved by developing a notion of “sectorial local-non-determinism (LND).” Both ideas may be of independent interest. We showcase sectorial LND further by exhibiting some arithmetic properties of standard Brownian motion; this completes the work initiated by Mountford (1988).


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Additional Information

Davar Khoshnevisan
Affiliation: Department of Mathematics, The University of Utah, 155 S. 1400 E., Salt Lake City, Utah 84112–0090
MR Author ID: 302544
Email: davar@math.utah.edu

Yimin Xiao
Affiliation: Department of Statistics and Probability, A-413 Wells Hall, Michigan State University, East Lansing, Michigan 48824
Email: xiao@stt.msu.edu

Keywords: Brownian sheet, image, Bessel–Riesz capacity, Hausdorff dimension, interior-point
Received by editor(s): September 12, 2004
Received by editor(s) in revised form: April 21, 2005
Published electronically: February 14, 2007
Additional Notes: This research was supported by a generous grant from the National Science Foundation
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.